#481226

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3481	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,21,26,23,25,36,41,22,27,35,42,20,32,34,45)(19,28,30,43,29,24,37,38,31,40,44,48,33,39,46,47);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,33)(21,30)(22,31)(23,29)(25,40)(26,38)(27,39)(28,32)(34,46)(35,44)(36,37)(41,48)(42,47)(43,45);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16);(3,10)(7,14)(11,16)(13,15)

Polycyclic presentation:

2^9|a^2=f,e^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,e^b=eg,d^c=dgi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287023, #72321994, #72322071
maximal normal subgroup of	#72287023, #72321994, #72322071

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388422);